(s1)
each \(u_i\in W^{1,\infty }(\overline{D})\)  ([1]}).
It follows that \(u_i\in C(\overline{D})\) , \(\omega _i\)  is open and \(p\in \omega _i\) 
implies \(m(p)=1\) . By standard regularity theory for elliptic
equations we also have that \(u_i\in C^\infty (\omega _i)\) ;
(s2)
each \(\omega _i\)  is connected and each \(\Gamma _{ij}\) 
is either empty or a connected arc starting from a point \(p_i\in \partial D\) 
([2]});
(s3)
\(u_i\)  is harmonic in \(\omega _i\) ; \(u_i-u_j\)  is harmonic
in \(D\setminus \cup _{h\ne i,j}\overline{\omega }_h\) 
[1]};
(s4)
if \(p\in D\)  satisfies \(m(p)=2\) , then
([1]})
\(\lim _{\omega _i\ni q\rightarrow p} \nabla u_i(q)=-\lim _{\omega _j\ni q\rightarrow p} \nabla u_j(q) ;\) 
(s5)
\(U\in W^{1,\infty }(\overline{D})\)  and if \(p\in D\)  satisfies
\(m(p)=2\) , then
\(|\nabla U(p)| = \lim _{\omega _i\ni q\rightarrow p} |\nabla u_i(q)|=\lim _{\omega _j\ni q\rightarrow p} |\nabla u_j(q)| \ne 0\) 
and the set \(\lbrace q:m(q)=2\rbrace \)  is locally a \(C^1\) -curve through
\(p\)  ending either at points with higher multiplicity,
or at the boundary \(\partial D\)  [1]};
(s6)
if \(p\in D\)  satisfies \(m(p)\ge 3\) , then
\(|\nabla U(q)|\rightarrow 0\) , as \(q\rightarrow p\)  [1]};
(s7)
the set \(\lbrace p\in D: m(p)\ge 3\rbrace \)  consists of a finite
number of points [1]};
(s8)
if \(p\in D\)  with \(m(p)=h\ge 3\)  then there
exists \(\theta _0\in (-\pi ,\pi ]\)  such that
\(U(r,\theta )=r^{h/2} \left| \cos \left( \frac{h}{2}(\theta +\theta _0)\right) \right|+o(r^{h/2})\) 
as \(r\rightarrow 0\) , where \((r,\theta )\)  is a system of polar coordinates
around \(p\)  [1]}.

